Linear Estimation for Approximately Linear Models

Abstract

An approximate linear model is proposed to allow for deviations from an underlying ideal linear model as follows: If, in standard notation, $Y = A\beta + \varepsilon$ is the ideal model then $Y = A\beta + r + \varepsilon$ where $|r_i| \leqq M_i$ for $M$ a given vector is an approximate linear model. The problem solved here is that of finding a linear estimate of a single linear function of $\beta$ which minimaxes mean square error in the approximate model. The estimate obtained may be the standard one from the ideal model, but in general it is not. The estimate is calculated as a solution to a set of nonlinear equations (generalizing the usual normal equations) and an algorithm is given for obtaining the solution.